Numerical boundary conditions for wide approximation finite difference

[amalak]

Hi,

I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is:

u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h²

or when discretized

u'' = -U_j-2 + 16U_j-1 -30U_j + 16U_j+1 -U_j+2 / 12h²

In addition to dirichlet boundary conditions (which are not troubling me to implement), I have to implement numerical boundary conditions for

U_-1 and U_N+1

The problem I'm encountering is I'm not sure what to try for these numerical boundary conditions (as in, I haven't a clue as to what may work). I have the scheme set up without those conditions, but that's not what I want. The only time I know U_-1 and U_N+1 come up are with Neumann boundary conditions, which I don't have.

Any help or pointers would be immensely appreciated, thank you.

 

[AlephZero]

Your Dirichlet boundary conditions are at $u_0$ and $u_N$, right?

I would create an unsymmetrical fourth order approximation for $u''_1$ in terms of $u_0, u_1, u_2, u_3, u_4$ and simiilarly for $u''_{N-1}$.

Then, you don't need $u_{-1}$ and $u_{N+1}$.

For example see section 11.3.4 of http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/lectfiniteDifference.pdf

 

[amalak]

Thank you very much for your response. That method seems to make more sense, I think, but I've been instructed to use U_-1 and U_N+1, but thank you again.

 

[AlephZero]

OK, so they probably want your numerical solution satisfy the differential equation at the boundary points $u_0$ and $u_N$, not just to satisfy the Dirichlet boundary conditions at the boundary points.

So, make finite difference approximations for the derivatives at $u_0$ using $u_{-1}, u_0,. u_1, \dots$ and plug them into the differential equation, and similarly at $u_N$. That will give you two more equations, so you have enough equations to solve for $u_{-1}, u_0,\dots, u_N, u_{N+1}$.

 

[alphy]

Dear researcher,

Thank you for sharing your work on using a wide 5 point stencil to solve a problem to fourth order accuracy. It seems like you have a well-defined mathematical approach to your problem, and I appreciate your attention to detail in ensuring accuracy.

In regards to your question about the numerical boundary conditions for U-1 and UN+1, I would suggest looking into using a ghost point approach. This involves extending your domain beyond the physical boundaries and using the values at these ghost points to approximate the values at the boundary points. This method has been used successfully in many numerical schemes and may be applicable in your case as well.

Another approach you could consider is using a higher order approximation scheme at the boundary points. This could involve using a wider stencil or incorporating additional terms in your finite difference equation to account for the boundary conditions.

I hope these suggestions are helpful in guiding you towards finding a suitable solution for your numerical boundary conditions. Keep up the good work and best of luck with your research!